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A ``nonlinear'' proof of Pitt's compactness theorem

Authors: M. Fabian and V. Zizler
Journal: Proc. Amer. Math. Soc. 131 (2003), 3693-3694
MSC (2000): Primary 46B25
Published electronically: July 9, 2003
MathSciNet review: 1998188
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Abstract: Using Stegall's variational principle, we present a simple proof of Pitt's theorem that bounded linear operators from $\ell_q$ into $\ell_p$ are compact for $1\le p<q<+\infty$.

References [Enhancements On Off] (What's this?)

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  • 2. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin, 1977. MR 58:17766
  • 3. R. R. Phelps, Convex functions, monotone operators, and differentiability, Lecture Notes in Math. No. 1364, 2nd Edition, Springer-Verlag, Berlin, 1993. MR 94f:46055
  • 4. Ch. Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Annalen 236 (1978), 171-176. MR 80a:46022
  • 5. Ch. Stegall, Optimization and differentiation in Banach spaces, Linear Algebra and Appl. 84 (1986), 191-211. MR 88a:49005

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Additional Information

M. Fabian
Affiliation: Mathematical Institute, Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic

V. Zizler
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Keywords: $\ell_p$ space, compact operator, variational principle
Received by editor(s): April 6, 2001
Published electronically: July 9, 2003
Additional Notes: Supported by grants GA ČR 201-98-1449, AV 1019003, and NSERC 7926
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society

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